Federated learning (FL) approaches for saddle point problems (SPP) have recently gained in popularity due to the critical role they play in machine learning (ML). Existing works mostly target smooth unconstrained objectives in Euclidean space, whereas ML problems often involve constraints or non-smooth regularization, which results in a need for composite optimization. Addressing these issues, we propose Federated Dual Extrapolation (FeDualEx), an extra-step primal-dual algorithm, which is the first of its kind that encompasses both saddle point optimization and composite objectives under the FL paradigm. Both the convergence analysis and the empirical evaluation demonstrate the effectiveness of FeDualEx in these challenging settings. In addition, even for the sequential version of FeDualEx, we provide rates for the stochastic composite saddle point setting which, to our knowledge, are not found in prior literature.
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The widespread use of maximum Jeffreys'-prior penalized likelihood in binomial-response generalized linear models, and in logistic regression, in particular, are supported by the results of Kosmidis and Firth (2021, Biometrika), who show that the resulting estimates are also always finite-valued, even in cases where the maximum likelihood estimates are not, which is a practical issue regardless of the size of the data set. In logistic regression, the implied adjusted score equations are formally bias-reducing in asymptotic frameworks with a fixed number of parameters and appear to deliver a substantial reduction in the persistent bias of the maximum likelihood estimator in high-dimensional settings where the number of parameters grows asymptotically linearly and slower than the number of observations. In this work, we develop and present two new variants of iteratively reweighted least squares for estimating generalized linear models with adjusted score equations for mean bias reduction and maximization of the likelihood penalized by a positive power of the Jeffreys-prior penalty, which eliminate the requirement of storing $O(n)$ quantities in memory, and can operate with data sets that exceed computer memory or even hard drive capacity. We achieve that through incremental QR decompositions, which enable IWLS iterations to have access only to data chunks of predetermined size. We assess the procedures through a real-data application with millions of observations, and in high-dimensional logistic regression, where a large-scale simulation experiment produces concrete evidence for the existence of a simple adjustment to the maximum Jeffreys'-penalized likelihood estimates that delivers high accuracy in terms of signal recovery even in cases where estimates from ML and other recently-proposed corrective methods do not exist.
The introduction of the European Union's (EU) set of comprehensive regulations relating to technology, the General Data Protection Regulation, grants EU citizens the right to explanations for automated decisions that have significant effects on their life. This poses a substantial challenge, as many of today's state-of-the-art algorithms are generally unexplainable black boxes. Simultaneously, we have seen an emergence of the fields of quantum computation and quantum AI. Due to the fickle nature of quantum information, the problem of explainability is amplified, as measuring a quantum system destroys the information. As a result, there is a need for post-hoc explanations for quantum AI algorithms. In the classical context, the cooperative game theory concept of the Shapley value has been adapted for post-hoc explanations. However, this approach does not translate to use in quantum computing trivially and can be exponentially difficult to implement if not handled with care. We propose a novel algorithm which reduces the problem of accurately estimating the Shapley values of a quantum algorithm into a far simpler problem of estimating the true average of a binomial distribution in polynomial time.
We derive an extension of the sequential homotopy method that allows for the application of inexact Krylov methods for the linear (double) saddle-point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off-diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.
Variance reduction is a crucial idea for Monte Carlo simulation and the stochastic Lanczos quadrature method is a dedicated method to approximate the trace of a matrix function. Inspired by their advantages, we combine these two techniques to approximate the log-determinant of large-scale symmetric positive definite matrices. Key questions to be answered for such a method are how to construct or choose an appropriate projection subspace and derive guaranteed theoretical analysis. This paper applies some probabilistic approaches including the projection-cost-preserving sketch and matrix concentration inequalities to construct a suboptimal subspace. Furthermore, we provide some insights on choosing design parameters in the underlying algorithm by deriving corresponding approximation error and probabilistic error estimations. Numerical experiments demonstrate our method's effectiveness and illustrate the quality of the derived error bounds.
We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the "sub-optimality" of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines--stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)--which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings.
For predictive modeling relying on Bayesian inversion, fully independent, or ``mean-field'', Gaussian distributions are often used as approximate probability density functions in variational inference since the number of variational parameters is twice the number of unknown model parameters. The resulting diagonal covariance structure coupled with unimodal behavior can be too restrictive when dealing with highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors in the form of Gaussian mixtures can capture any distribution to an arbitrary degree of accuracy while maintaining some analytical tractability. Variational inference with Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. Coupled with the existence of multiple local minima due to nonconvex trends in the loss functions often associated with variational inference, these challenges motivate the need for robust initialization procedures to improve the performance and scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with an optimization stage in model parameter space in which local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace method. Finally, the mixture weights are determined through constrained least squares regression. Robustness and scalability are demonstrated using synthetic tests. The methodology is applied to an inversion problem in structural dynamics involving unknown viscous damping coefficients.
This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components separately, linearizing only the smooth parts. This provides us with new generalizations of the classical Frank-Wolfe method and the Conditional Gradient Sliding algorithm, that cater to a subclass of non-differentiable problems. Our algorithms rely on a stronger version of the linear minimization oracle, which can be efficiently implemented in several practical applications. We provide the basic version of our method with an affine-invariant analysis and prove global convergence rates for both convex and non-convex objectives. Furthermore, in the convex case, we propose an accelerated method with correspondingly improved complexity. Finally, we provide illustrative experiments to support our theoretical results.
The number of modes in a probability density function is representative of the model's complexity and can also be viewed as the number of existing subpopulations. Despite its relevance, little research has been devoted to its estimation. Focusing on the univariate setting, we propose a novel approach targeting prediction accuracy inspired by some overlooked aspects of the problem. We argue for the need for structure in the solutions, the subjective and uncertain nature of modes, and the convenience of a holistic view blending global and local density properties. Our method builds upon a combination of flexible kernel estimators and parsimonious compositional splines. Feature exploration, model selection and mode testing are implemented in the Bayesian inference paradigm, providing soft solutions and allowing to incorporate expert judgement in the process. The usefulness of our proposal is illustrated through a case study in sports analytics, showcasing multiple companion visualisation tools. A thorough simulation study demonstrates that traditional modality-driven approaches paradoxically struggle to provide accurate results. In this context, our method emerges as a top-tier alternative offering innovative solutions for analysts.
As digital transformation continues, enterprises are generating, managing, and storing vast amounts of data, while artificial intelligence technology is rapidly advancing. However, it brings challenges in information security and data security. Data security refers to the protection of digital information from unauthorized access, damage, theft, etc. throughout its entire life cycle. With the promulgation and implementation of data security laws and the emphasis on data security and data privacy by organizations and users, Privacy-preserving technology represented by federated learning has a wide range of application scenarios. Federated learning is a distributed machine learning computing framework that allows multiple subjects to train joint models without sharing data to protect data privacy and solve the problem of data islands. However, the data among multiple subjects are independent of each other, and the data differences in quality may cause fairness issues in federated learning modeling, such as data bias among multiple subjects, resulting in biased and discriminatory models. Therefore, we propose DBFed, a debiasing federated learning framework based on domain-independent, which mitigates model bias by explicitly encoding sensitive attributes during client-side training. This paper conducts experiments on three real datasets and uses five evaluation metrics of accuracy and fairness to quantify the effect of the model. Most metrics of DBFed exceed those of the other three comparative methods, fully demonstrating the debiasing effect of DBFed.
As data are increasingly being stored in different silos and societies becoming more aware of data privacy issues, the traditional centralized training of artificial intelligence (AI) models is facing efficiency and privacy challenges. Recently, federated learning (FL) has emerged as an alternative solution and continue to thrive in this new reality. Existing FL protocol design has been shown to be vulnerable to adversaries within or outside of the system, compromising data privacy and system robustness. Besides training powerful global models, it is of paramount importance to design FL systems that have privacy guarantees and are resistant to different types of adversaries. In this paper, we conduct the first comprehensive survey on this topic. Through a concise introduction to the concept of FL, and a unique taxonomy covering: 1) threat models; 2) poisoning attacks and defenses against robustness; 3) inference attacks and defenses against privacy, we provide an accessible review of this important topic. We highlight the intuitions, key techniques as well as fundamental assumptions adopted by various attacks and defenses. Finally, we discuss promising future research directions towards robust and privacy-preserving federated learning.
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